📐 Estimation in Statistics

Last Updated: Jan 2026

Estimation is the process of using sample data to estimate an unknown population parameter.

Population parameters like:

  • Mean (μ)
  • Proportion (p)
  • Variance (σ²)

are usually unknown, so we estimate them using samples.

🗣 Hinglish Tip: Estimation = sample ke basis par population ka value guess karna (scientific way me)

Types of Estimation

There are two main types:

  1. Point Estimation
  2. Interval Estimation (Confidence Interval)

Point Estimation

Point estimation gives a single numerical value as an estimate of a population parameter.

Common Point Estimators

Population ParameterEstimatorNotation
Population MeanSample Meanx̄
Population VarianceSample Variances²
Population ProportionSample Proportionp̂

Example: Point Estimation of Mean

  • Sample data: 10, 12, 14, 16, 18
  • Sample size: n = 5

Solution

x̄ = (10 + 12 + 14 + 16 + 18) / 5
x̄ = 70 / 5
x̄ = 14

✔ Point estimate of population mean = 14

⚠ Limitation:

  • No idea about accuracy or reliability
  • No range given

Interval Estimation (Confidence Interval)

Interval estimation gives a range of values within which the population parameter is likely to lie.

This range is called a Confidence Interval (CI).

Confidence Interval (CI)

A confidence interval is written as:

Lower Limit  <  Parameter  <  Upper Limit

Common confidence levels:

  • 90%
  • 95%
  • 99%

🗣 Hinglish Tip: 95% CI = hume 95% confidence hai ki true mean is range ke andar hoga

Confidence Interval for Mean (σ known)

Formula

CI = x̄ ± Z (σ / √n)

Where:

  • x̄ = sample mean
  • σ = population standard deviation
  • n = sample size
  • Z = Z-value (from normal table)

Z-values (Common)

Confidence LevelZ-value
90%1.645
95%1.96
99%2.58

Example

  • Sample mean: x̄ = 50
  • Population standard deviation: σ = 10
  • Sample size: n = 100
  • Confidence level: 95%

Step 1: Identify Z-value

From table:

Z = 1.96

Step 2: Calculate Standard Error

σ / √n = 10 / √100 = 10 / 10 = 1

Step 3: Calculate Margin of Error

Margin of Error = Z × Standard Error
Margin of Error = 1.96 × 1 = 1.96

Step 4: Construct Confidence Interval

Lower Limit = 50 − 1.96 = 48.04
Upper Limit = 50 + 1.96 = 51.96

✅ Final Answer

48.04 < μ < 51.96

✔ We are 95% confident that population mean lies between 48.04 and 51.96

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